Question

divide using polynomial long division or synthetic division.\\((x^3 + x^2 + 3x - 4) \div (x^2 + 2x + 1) = \square\\)

Explanation:

Step1: Divide leading terms

$\frac{x^3}{x^2} = x$

Step2: Multiply divisor by $x$

$x(x^2 + 2x + 1) = x^3 + 2x^2 + x$

Step3: Subtract from dividend

$(x^3 + x^2 + 3x - 4) - (x^3 + 2x^2 + x) = -x^2 + 2x - 4$

Step4: Divide new leading terms

$\frac{-x^2}{x^2} = -1$

Step5: Multiply divisor by $-1$

$-1(x^2 + 2x + 1) = -x^2 - 2x - 1$

Step6: Subtract to get remainder

$(-x^2 + 2x - 4) - (-x^2 - 2x - 1) = 4x - 3$

Step7: Write final form

Quotient plus remainder over divisor

Answer:

$x - 1 + \frac{4x - 3}{x^2 + 2x + 1}$